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Suppose V is a vector space over F, dim V = n, let T be a...

Suppose V is a vector space over F, dim V = n, let T be a linear transformation on V.

1. If T has an irreducible characterisctic polynomial over F, prove that {0} and V are the only T-invariant subspaces of V.

2. If the characteristic polynomial of T = g(t) h(t) for some polynomials g(t) and h(t) of degree < n , prove that V has a T-invariant subspace W such that 0 < dim W < n

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